I am System Admin and create Permission sets with Manage Public Lightning Email Templates and Create Folders for Lightning Email Templates this permission.
Yes, the Salesforce help page on this item is unclear, to say the least. It mentions the permission, which you need to add via a permission set, by the way, you cannot add it directly to the sysadmin profile.
The help page states:
From the Email Templates page
So you go to Setup, and there you will find Classic email templates and Lightning email templates, but not Email templates. Fair enough, you go to Lightning email templates, only to find the switch that you see in the question. Ctrl-F5, logging in and out, restarting browser, nope, only the switch and nothing else.
And then you start googling some more and find out that Email templates is not in Setup, but in All Items in the All Launcher. This is not intuitive at all and the help page is not specific enough. I have posted feedback on the help page, let's hope they fix it.